We prove the existence of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers of a radial coordinate.

We prove that for any vacuum, maximal, asymptotically flat, axisymmetric initial data for Einstein equations close to extreme Kerr data, the inequality √ J m is satisfied, where m and J are the total mass and angular momentum of the data. The proof consists in showing that extreme Kerr is a local minimum of the mass.

We evolve equal-mass, equal-spin black-hole binaries with specific spins of a/mH ∼ 0.925, the highest spins simulated thus far and nearly the largest possible for Bowen-York black holes, in a set of configurations with the spins counter-aligned and pointing in the orbital plane, which maximizes the recoil velocities of the merger remnant, as well as a configuration where the two spins point in the same direction as the orbital angular momentum, which maximizes the orbital hang-up effect and remnant spin. The coordinate radii of the individual apparent horizons in these cases are very small and the simulations require very high central resolutions (h ∼ M/320). We find that these highly spinning holes reach a maximum recoil velocity of ∼ 3300 km s −1 (the largest simulated so far) and, for the hangup configuration, a remnant spin of a/mH ∼ 0.922. These results are consistent with our previous predictions for the maximum recoil velocity of ∼ 4000 km s −1 and remnant spin; the latter reinforcing the prediction that cosmic censorship is not violated by merging highlyspinning black-hole binaries. We also numerically solve the initial data for, and evolve, a single maximal-Bowen-York-spin black hole, and confirm that the 3-metric has an O(r −2 ) singularity at the puncture, rather than the usual O(r −4 ) singularity seen for non-maximal spins.